# Langevin dynamcs MCMC

August 17, 2020 — October 1, 2022

## 1 Langevin dynamics

## 2 Metropolis-adjusted Langevin algorithm (MALA)

## 3 Continuous time

See log concave distributions for a family of distributions where this works especially well because implcit (more nearly continuous-time exact) solutions are available Hodgkinson, Salomone, and Roosta (2019).

Left-field, Max Raginsky, Sampling Using Diffusion Processes, from Langevin to Schrödinger:

the Langevin process gives only approximate samples from \(\mu\). I would like to discuss an alternative approach that uses diffusion processes to obtain exact samples in finite time. This approach is based on ideas that appeared in two papers from the 1930s by Erwin Schrödinger in the context of physics, and is now referred to as the Schrödinger bridge problem.

## 4 Annealed

TBC Jolicoeur-Martineau et al. (2022);Song and Ermon (2020a);Song and Ermon (2020b).

Yang Song, Generative Modeling by Estimating Gradients of the Data Distribution

## 5 Incoming

Holden Lee, Andrej Risteski introduce the connection between log-concavity and convex optimisation.

\[ x_{t+\eta} = x_t - \eta \nabla f(x_t) + \sqrt{2\eta}\xi_t,\quad \xi_t\sim N(0,I). \]

## 6 References

*Proceedings of the 32nd International Conference on Neural Information Processing Systems*. NIPS’18.

*arXiv:1704.04752 [Math, Stat]*.

*arXiv:1605.01559 [Math, Stat]*.

*SIAM Journal on Applied Dynamical Systems*.

*arXiv:1812.00793 [Cs, Math, Stat]*.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

*Journal of the Royal Statistical Society: Series B (Methodological)*.

*arXiv:1903.12322 [Cs, Stat]*.

*Proceedings of the 36th International Conference on Machine Learning*.

*arXiv:1610.00781 [Math, Stat]*.

*Nuclear Physics B*.

*Statistics & Probability Letters*.

*Advances in Neural Information Processing Systems*. NIPS’15.

*Advances In Neural Information Processing Systems*.

*Advances In Neural Information Processing Systems*.

*Proceedings of the 28th International Conference on International Conference on Machine Learning*. ICML’11.

*Statistics & Probability Letters*.

*International Conference on Artificial Intelligence and Statistics*.